Thursday, November 13, 2014

Section 6-5: Congruent Figures (Day 63)

Yesterday I tutored my geometry student. Chapter 2 of the Glencoe text is very similar to the same chapter in the U of Chicago. But Section 2-1 of the Glencoe text is on "Inductive Reasoning and Conjecture," which does not correspond exactly to a section in the U of Chicago. Sometimes it's fun to have the students look for patterns and guess what is next in a sequence. It may be a sequence of numbers, letters, colors, or shapes. One of my favorite sequences (not mentioned in Glencoe, but in yet another geometry text that I've seen) begins George, John, Thomas, James. The sequence is the first names of the U.S. Presidents. (A similar sequence begins George, Thomas, Abe, Alexander -- the faces that appear on U.S. money.)

I showed my student my worksheet from this blog, for Section 2-2, which mentions some of the material in the first three sections of Glencoe Chapter 2. He had no trouble with most of it -- but I noticed that he preferred "If |x| = 3, then x = 3" as a statement to find a counterexample, instead of what I had written, "If x^2 = 9, then...."

Today, meanwhile, I subbed in an eighth grade English class. But one student arrived late -- because he had to travel from the high school, where he attended a Common Core Integrated Math class. Some opponents of Common Core often point out that it is more difficult to accelerate and take higher math classes under the Common Core. But in this case, his needing to travel would have occurred under a pre-Core math -- only it would have been geometry instead of integrated math. It's always been difficult for a student in the highest grade level at elementary or middle school to accelerate. I do have more to say about the Common Core debate in this post, but not about acceleration. Instead, I want to discuss the way that Common Core teaches congruence.

Section 6-5 of the U of Chicago text is on congruent figures. Congruence is one of the most important concepts in all of geometry, especially Common Core Geometry.

As I mentioned many times on this blog, the word congruent is defined very differently in Common Core Geometry than under previous standards. We all know what it means for two segments to be congruent -- that is, that they have equal length -- or for two angles to be congruent -- that is, that they have equal measure. The new definition of congruent appears to be original to the Common Core, and yet, it isn't. Years before the Core, the U of Chicago text used the following definition of congruent -- indeed, it is mainly because of this definition that I chose the U of Chicago as the textbook on which this blog is based:

Two figures, F and G, are congruent figures [...] if and only if G is the image of F under a translation, a reflection, a rotation, or any composite of these.

And there we have it -- this definition of congruent predates Common Core. But many opponents of Common Core do not like this new definition. One such opponent is Ze'ev Wurman, a member of the commission here in California that reviewed the Common Core standards. Although his views are posted at several websites, one of the best Wurman articles I found is at this link:

Skipping down to the discussion of the math standards -- since, as Wurman himself points out, math is his area of expertise -- the author begins with some elementary school standards. For example, Wurman gives this standard:

1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Wurman states that this standard should have stopped after the first sentence. Instead, it goes on to prescribe some nonstandard algorithms for addition. I already discussed much of this back during the week between Days 51 and 52. Since much of what Wurman writes about grades 1-3 echo what I wrote about the lower grades, I am in full agreement with Wurman for the lower grades.

But then Wurman moves on to Common Core Geometry. Here's what he writes about it:

A true content standard would simply say “Students prove triangle congruence” or, perhaps, “Students understand triangle congruence,” leaving the method of instruction to the teacher. Instead, Common Core not only dictates how to teach congruence, it insists on a specific experimental method of instruction that has an established a track record of failure where it was invented [...]
[emphasis Wurman's]

He then gives a link to a PDF file about the Russian mathematician and scientist A.N. Kolmogorov, whom the PDF credits as the creator of geometry based on transformations. Wurman implies that this geometry was tried out in Russia (i.e., the Soviet Union) and was a big failure.

The link between Kolmogorov's axioms and Common Core, to me, is uncertain -- but it is possible that both U of Chicago and Common Core derived their ideas from Kolmogorov.

Why should we use the transformation approach? In pre-Core Geometry, we must define the word congruent three times -- first for segments, then for angles, and finally for figures. But in some ways, this is an ad hoc approach. In the Common Core, we only define congruent once, and it applies to segments, angles, and figures all at once.

In the Common Core, congruent means "identical up to isometry" (and later on, we see that similar is defined as "identical up to a similarity transformation"). There are many concepts in college-level mathematics that are defined similarly -- such as topologically equivalent ("identical up to homeomorphism"), equinumerous ("identical up to bijection"), and so on. Furthermore, the Lebesgue measure of a set is defined so that two sets such that there is an isometry mapping one to the other have the same measure.

So in some ways, the Common Core definition is more rigorous than the pre-Core definitions. Also, in some ways, the Common Core definition predates Kolmogorov by a wide margin -- Euclid himself used it as the Principle of Superposition in his proof of SAS (Proposition I.4).

In Hilbert's formulation of Euclid's axioms, congruence is a primitive notion -- that is, it is undefined just as point, line, and plane are. Actually, it's two undefined terms, since Hilbert considers segment and angle congruence separately. As I mentioned before, we can't define an undefined term, but instead we know what it means through the use of axioms or postulates. Hilbert provides six axioms of congruence -- these cover the Equivalence Properties and Segment and Angle Congruence Theorems as given in this section, some of the Point-Line-Plane and Angle Measure Postulates, and SAS. We notice that Hilbert's congruence is completely nonmetric -- there is no notion of distance or angle measure anywhere.

So which formulation should we use? This is a Common Core site and so I use the Common Core definition of congruence, but in the long run, which is best for the students? The usual guiding principles is that if a concept is easy for the students to understand and leads to a higher concept, then the students should learn how to prove it. But if the lower concept is difficult for the students, it should be made into a postulate and not proved in class.

So we can see a full continuum, from more proofs to more postulates:

  • Common Core: SAS, ASA, SSS all proved (using transformations)
  • Hilbert (supported by David Joyce): SAS postulated, ASA, SSS proved (using SAS)
  • Status quo: SAS, ASA, SSS all postulated (most texts)
  • Minimalist: SAS, ASA, SSS not mentioned (isosceles/parallelogram properties postulated)
The argument from Wurman and other Common Core opponents is that proving SAS, ASA, SSS from transformations only confused students (which would be the reason why this would have been a big failure in the Soviet Union) and that they should be assumed as postulates. But then, we wonder, why not go one step further and state that all proofs confuse students, so that all proofs involving SAS, ASA, SSS should be dropped, and the properties of triangles and parallelograms assumed? Why is the status quo, where SAS, ASA, SSS are assumed and used in proofs, exactly the right level of complexity for the students?

Well, this is what I hope to find out through this blog. It could be that these Core opponents are correct, and that the status quo level of complexity is exactly appropriate for high school students taking geometry. To me, this is not as clear-cut as elementary math, where the standard algorithms for addition and subtraction are clearly superior to the nonstandard algorithms. This is the reason that I've summarized my view of K-3 math, but not high school math yet.

As for the other theorems proved in this chapter, the Equivalence Properties of Congruence is proved in a way that is standard for many types of transformations -- by using the identity, inverse, and composite functions. The Segment and Angle Congruence Theorems are proved using reflections only, since the text states (in the "Shorter Form" of the definition of congruence) that only reflections, or a composite thereof, are needed to establish congruence. But sometimes it's easier for students to visualize other transformations -- for example, in the Segment Congruence Theorem, one can simply translate X to Z, so that X' and Z coincide. Then one can rotate W' to Y, so that W" and Y coincide. In the text, both of these are reflections instead.

Notice that this section, 6-5, is the first lesson in which the word congruent appears. The U of Chicago text is careful to use phrases such as "of equal length (measure)" instead of congruent.

I've mentioned before that many people -- both teachers and subjects -- use the words equal and congruent interchangeably. There are two distinctions to make -- one is that numbers (including lengths and angle measures) are equal, while segments and angles are congruent. The other is that we don't know that any figures are congruent until we know of an isometry mapping one to the other, which the Segment and Angle Congruence Theorems provide.

In this course, the latter distinction has priority. I admit that I myself have called angles "equal" (when it's their measures that are equal) on this blog -- because I don't want to call them "congruent" until reaching the Segment and Angle Congruence Theorems. I am especially guilty of this when I write phrases such as "Angle A = Angle B" because it's so much easier than trying to write an angle symbol in ASCII. Occasionally, I would underline a slash: m /  A = m /  B is the best I can do. Of course, I can't really draw a congruent sign at all, unless I write ~= and you just imagine that the tilde is directly above the equal sign.

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